# Differences between R and Patsy formulas¶

Patsy has a very high degree of compatibility with R. Almost any formula you would use in R will also work in Patsy – with a few caveats.

Note

All R quirks described herein were last verified with R 2.15.0.

Differences from R:

• Most obviously, we both support using arbitrary code to perform variable transformations, but in Patsy this code is written in Python, not R.

• Patsy has no %in%. In R, a %in% b is identical to b:a. Patsy only supports the b:a version of this syntax.

• In Patsy, only ** can be used for exponentiation. In R, both ^ and ** can be used for exponentiation, i.e., you can write either (a + b)^2 or (a + b)**2. In Patsy (as in Python generally), only ** indicates exponentiation; ^ is ignored by the parser (and if present, will be interpreted as a call to the Python binary XOR operator).

• In Patsy, the left-hand side of a formula uses the same evaluation rules as the right-hand side. In R, the left hand side is treated as R code, so a formula like y1 + y2 ~ x1 + x2 actually regresses the sum of y1 and y2 onto the set of predictors x1 and x2. In Patsy, the only difference between the left-hand side and the right-hand side is that there is no automatic intercept added to the left-hand side. (In this regard Patsy is similar to the R enhanced formula package Formula.)

• Patsy produces a different column ordering for formulas involving numeric predictors. In R, there are two rules for term ordering: first, lower-order interactions are sorted before higher-order interactions, and second, interactions of the same order are listed in whatever order they appeared in the formula. In Patsy, we add another rule: terms are first grouped together based on which numeric factors they include. Then within each group, we use the same ordering as R.

• Patsy has more rigorous handling of the presence or absence of the intercept term. In R, the rules for when deciding whether to include an intercept are somewhat idiosyncratic and can ignore things like parentheses. To understand the difference, first consider the formula a + (b - a). In both Patsy and R, we first evaluate the (b - a) part; since there is no a term to remove, this simplifies to just b. We then evaluate a + b: the end result is a model which contains an a term in it.

Now consider the formula 1 + (b - 1). In Patsy, this is analogous to the case above: first (b - 1) is reduced to just b, and then 1 + b produces a model with intercept included. In R, the parentheses are ignored, and 1 + (b - 1) gives a model that does not include the intercept.

This can be slightly more confusing when it comes to the implicit intercept term. In Patsy, this is handled exactly as if the right-hand side of each formula has an invisible "1 +" inserted at the beginning. Therefore in Patsy, these formulas are different:

# Python:
dmatrices("y ~ b - 1")   # equivalent to 1 + b - 1: no intercept
dmatrices("y ~ (b - 1)") # equivalent to 1 + (b - 1): has intercept


In R, these two formulas are equivalent.

• Patsy has a more accurate algorithm for deciding whether to use a full- or reduced-rank coding scheme for categorical factors. There are two situations in which R’s coding algorithm for categorical variables can become confused and produce over- or under-specified model matrices. Patsy, so far as we are aware, produces correctly specified matrices in all cases. It’s unlikely that you’ll run into these in actual usage, but they’re worth mentioning. To illustrate, let’s define a and b as categorical predictors, each with 2 levels:

# R:
> a <- factor(c("a1", "a1", "a2", "a2"))
> b <- factor(c("b1", "b2", "b1", "b2"))


The first problem occurs for formulas like 1 + a:b. This produces a model matrix with rank 4, just like many other formulas that include a:b, such as 0 + a:b, 1 + a + a:b, and a*b:

# R:
> qr(model.matrix(~ 1 + a:b))\$rank
[1] 4


However, the matrix produced for this formula has 5 columns, meaning that it contains redundant overspecification:

# R:
> mat <- model.matrix(~ 1 + a:b)
> ncol(mat)
[1] 5


The underlying problem is that R’s algorithm does not pay attention to ‘non-local’ redundancies – it will adjust its coding to avoid a redundancy between two terms of degree-n, or a term of degree-n and one of degree-(n+1), but it is blind to a redundancy between a term of degree-n and one of degree-(n+2), as we have here.

Patsy’s algorithm has no such limitation:

# Python:
In [1]: a = ["a1", "a1", "a2", "a2"]

In [2]: b = ["b1", "b2", "b1", "b2"]

In [3]: mat = dmatrix("1 + a:b")

In [4]: mat.shape[1]
Out[4]: 4


To produce this result, it codes a:b uses the same columns that would be used to code b + a:b in the formula "1 + b + a:b".

The second problem occurs for formulas involving numeric predictors. Effectively, when determining coding schemes, R assumes that all factors are categorical. So for the formula 0 + a:c + a:b, R will notice that if it used a full-rank coding for the c and b factors, then both terms would be collinear with a, and thus each other. Therefore, it encodes c with a full-rank encoding, and uses a reduced-rank encoding for b. (And the 0 + lets it avoid the previous bug.) So far, so good.

But now consider the formula 0 + a:x + a:b, where x is numeric. Here, a:x and a:b will not be collinear, even if we do use a full-rank encoding for b. Therefore, we should use a full-rank encoding for b, and produce a model matrix with 6 columns. But in fact, R gives us only 4:

# R:
> x <- c(1, 2, 3, 4)
> mat <- model.matrix(~ 0 + a:x + a:b)
> ncol(mat)
[1] 4


The problem is that it cannot tell the difference between 0 + a:x + a:b and 0 + a:c + a:b: it uses the same coding for both, whether it’s appropriate or not.

(The alert reader might wonder whether this bug could be triggered by a simpler formula, like 0 + x + b. It turns out that R’s code do_modelmatrix function has a special-case where for first-order interactions only, it will peek at the type of the data before deciding on a coding scheme.)

Patsy always checks whether each factor is categorical or numeric before it makes coding decisions, and thus handles this case correctly:

# Python:
In [5]: x = [1, 2, 3, 4]

In [6]: mat = dmatrix("0 + a:x + a:b")

In [7]: mat.shape[1]
Out[7]: 6